But the motion of the bob, under the influence of the pull of an elastic band where the strain was always proportional to the displacement, would, as we have seen, be harmonic motion, and performed in equal times whatever the extent of the swing. Whence then we conclude that if the swings of a pendulum are not too big, say not exceeding two and a half inches each way, the motion may be considered harmonic motion, and the swings will be made in equal times whether they are large or small ones. In other words, a clock with a 39-1/7 inch pendulum and side swing on each side if not over two inches will keep time, whatever the arc of swing may be. This may be verified experimentally. Take a pendulum of wood 39-1/7 inches long, and affix to its end a bob of 10 lbs. weight. The pendulum will swing once in each second. To pull it aside two inches we should want a weight such that its moment about the point of support was equal to the moment of the force of gravity acting on the bob, about the point of support. In other words, the weight required × 39-1/7 inches = 10 lbs. × 2 inches. Whence the weight required = 1/2 lb. (nearly). Now fix a similar pendulum _A B_ 39-1/7 inches long, horizontally, with a weight _B_ of 10 lbs. on it. Fasten it to a vertical shaft _C D_, with a tie rod of wire or string _A B_ so as to keep it up, and attach to each side of the rod _A B_ elastic threads _E F_ and _E G_. Let these threads be tied on at such a point that when _B_ is pulled aside two inches the force tending to bring it back to rest is half a pound. Then if set vibrating the rod will swing backwards and forwards in equal times, no matter how big, the arc of vibration (provided the arc is kept small), and the time of oscillation will be that of a pendulum, namely, one swing in a second. In fact, whether you put _A B_ vertically and let it swing on the pivots _C_ and _D_ by the force of gravity, or put it horizontally, and thus prevent gravity acting on it, but make it swing under the accelerating influence of a pair of elastic bands so arranged as to be equivalent to gravity, in each case it will swing in seconds. [Illustration: FIG. 38.] It is this curious property of the circle that makes the vertical force of gravity on a pendulum pull it as though it were a horizontally acting elastic band; that is the reason why a pendulum is equal-time-swinging, or, as it is called, isochronous, from two Greek words that mean “the same” and “time.” But it must be remembered that this equal swinging is only approximate, and only true when the arc of vibration is small. Here then we have a proof which shows us that the pendulum of a clock and the balance wheel of a watch depend on exactly the same principles. They are each an example of harmonic motion. The next question that arises is whether the weight of the pendulum has any influence upon the time of its vibration.